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ON THE ULTIMATE FINENESS OF A DISPERSIONE.B. NAUMAN a , IVO ROUŠAR a & ASHIM DUTTA aa The Isermann Department of Chemical Engineering , Rensselaer Polytechnic Institute ,Troy, NY, 12180-3590Published online: 15 Oct 2007.

To cite this article: E.B. NAUMAN , IVO ROUŠAR & ASHIM DUTTA (1991) ON THE ULTIMATE FINENESS OF A DISPERSION,Chemical Engineering Communications, 105:1, 61-75, DOI: 10.1080/00986449108911518

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Chem. Eng. Comm. 1991, Vol. 105, pp. 61-75 Reprints available directly from the publisher. Photocopying permitted by license only. 0 1991 Gordon and Breach Science Publishers S.A. Printed in the United States of America

ON THE ULTIMATE FINENESS OF A DISPERSION

E.B. NAUMAN, IVO ROUSARP, and ASHIM D U l T A

The Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute,

Troy, NY 121 80-3590

(Received September 11, 1990; in final form April 4, 1991)

Equilibrium composition and pressure profiles are determined for spherical drops within a binary. Landau-Ginzburg fluid that has undergone phase separation. There is a smallest possible drop size, below which a drop would redissolve to form a homogeneous system. This minimum drop size has a radius of - where K is the gradient energy parameter and g(6) is the Gibbs free energy of mixing evaluated at the system average concentration. The minimum drop size at equilibrium is approximately equal to the minimum size for growth by spinodal decomposition as predicted from linearized Cahn-Hilliard theory. It represents a practical limit on the ultimate fineness of polymer-in-polymer microdisper- sions. The minimum drop radius will be on the order of 0.02 microns for typical, high molecular weight polymers.

The surface tension is calculated for systems of finite extent in both radial and spherical coordinates. It vanishes when the size of the system is less than the minimum size needed for bifurcation into two phases. The pressure distribution within spherical drops is calculated using a differential form of the Young-Laplace equation. The classical result,

overpredicts the internal pressure for small drop sizes although this is partially due to the ambiguity in specifying the radius of small drops. KEYWORDS Dispersion Phase separation Phase equilibrium Landau-Ginzburg

Spinodal decomposition Drop size.

INTRODUCTION

Engineers frequently seek to disperse one liquid within a matrix of a second, incompatible liquid. It is not widely recognized that there is a smallest possible size, below which a dispersed phase cannot exist at equilibrium. Subdivision of a drop below the critical size might be achievable by mechanical means, but the resultant droplets are unstable and would dissolve into the matrix phase. The smallest possible size is on the order of a few tens of angstroms for ordinary liquids. Thus, it does not represent a practical limit except, perhaps, in confined systems such as zeolite pores. For polymer molecules, however, the critical size is

t Present address: Deprtment of Equipment Design for Process Industries, Czech Technical University, Suchbatarova 4, 116 07 Prague, Czechoslovakia.

61

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62 DR E.B. NAUMAN et al.

a few hundred angstroms and represents a significant limit on the achievable fineness of a polymer-in-polymer microdispersion.

The present work is specifically applicable to polymers but also continues a more general study of phase structures and phase transitions in binary liquid systems. A previous paper (Nauman and Balsara, 1988) calculated equilibrium composition distributions in Landau-Ginzburg fluids for one dimensional systems of finite size. The present paper extends the theory to spherical drops. Equilibrium composition distributions are calculated and the minimum possible size of a drop at equilibrium is determined. Surface tensions and the pressure distribution inside spherical drops are also found.

The free energy of mixing is assumed to be given by the Landau-Ginzburg functional:

Here, a = a(r, 0, $) is the mole fraction of component A and G = g + (~/2)(Va) ' is the free energy per unit volume. The ordinary or macroscopic free energy per unit volume is g = g(a). Equation (1) is the lowest order approxima- tion to the situation where the free energy depends not just on local concentra- tions but on the spatial distribution of concentrations. More complicated and presumably more accurate functionals are available for small molecules, but Eq. (1) seems fairly well suited for polymers. The macroscopic free energy, g ( a ) , is reasonably represented by Flory-Huggins theory, and the parameter K has been estimated in considerable detail (see Ariyapadi and Nauman, 1990).

As formulated above, g has units of energy/volume = pressure and K has units of force. It is convenient to divide both sides of Eq. (1) by pR,T where p is the molar density, R, is the gas constant and T is the absolute temperature. In most of what follows, we shall assume that this has been done and that Eq. (1) is written in terms of variables scaled by pR,T. Thus g is dimensionless and K has units of area. For numerical examples, we suppose that the macroscopic system has a free energy given by regular solution theory. The scaled g is

where x is an interaction coefficient that accounts for a nonideal enthalpy of mixing and where a is the mole fraction of one component. The Flory-Huggins equation is identical to Eq. (1) when a is replaced by the volume fraction of one polymer and x = mx,, where m is the degree of polymerization and xn, is the Flory-Huggins interaction parameter. When x = 0, the solution is ideal and will remain homogeneous. If x > 2, phase separation will occur in an infinitely large (i.e., macroscopic) system; but, if K >O, will not occur in systems smaller than some critical size. For a binary polymer-polymer system, K can be approximated as

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ULTIMATE FINENESS O F A DISPERSION 63

where Rm and R, are the radii of gyration of the two types of polymer molecules and x is the interaction parameter between them. Here we assume the polymers to have similar degrees of polymerization, rn, and similar radii of gyration, RG.

CONCENTRATION PROFILES IN A SPHERE

Consider a spherical container of radius R within which is a two phase liquid mixture governed by Eq. (1). We suppose a drop of one phase, rich in component A, is centered within a spherical annulus of the second phase, lean in component A. The composition at the center is a(r = 0 ) =a , . The edge of the drop can be defined as the inflection point, R,, where d2a/dr2 = 0 and where a(R i ) = ai (other definitions are possible). The composition at the wall of the container is a(R) =a , . For this case of spherical symmetry, Eq. (1) becomes

The composition profile, a(r ) , is determined by minimizing GI,,,, subject to the material balance constraint that

R i=el ar2dr v 0

Minimization of Eq. (4) subject to Eq. (5 ) gives equation of the form

(5)

rise to an Euler-Lagrange

where g l (a ) = dglda and y is a constant. The boundary conditions associated with Eq. (6) are

da - = 0 at r = O and r = R dr

It should be noted that the postulated sphere-within-a-sphere geometry represents a local minimum rather than the global minimum of Eq. (1). The container wall is assumed to be nonenergetic, and thus the global minimum occurs when both phases are in contact with the wall.

The postulated geometry approximates freely suspended drops of one phase in another. We can suppose the continuous phase has composition a, so that a( r ) is continuous at r = R . Although not at equilibrium, such systems can be fairly stable. In what follows, we shall assume that coarsening mechanisms such as Brownian agglomeration, settling and Ostwald ripening are sufficiently slow that the equilibrium composition profile, a(r ) , governed by Eq. (6) can be closely approximated.

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64 DR E.B. NAUMAN et al.

We turn now to the problem of calculating a(r). A previous paper (Nauman and Balsara, 1988) used generalized chemical potentials to aid in the calculations. This concept is still valid theoretically but is less useful in the present geometry (see Appendix 1). Equation (6) can be replaced by a set of simultaneous, first order equations

where w = r l f i . This set is integrated beginning at w = 0. The initial conditions are a(o) = a, and b(o) = 0. Values for a, and y are assumed. Integration continues until daldw = 0 at w = W = RIG. The assumed value for y can be adjusted to give a desired value for the system average concentration, 6. The gradient energy parameter is calculated using

Figure 1 gives sample results for a regular solution with x = 3. The numerical solution of Eqs. (8) give rise to computational difficulties that

have an underlying physical interpretation. The concentration at the center.

0 I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DIMENSIONLESS RADIUS, r/R FIGURE I Equilibrium concentration profiles for a Landau-Ginzburgh fluid: ti = 0.5, x = 3.

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CENTRAL CONCENTRATION, a, FIGURE 2 Asymmetry in central and outer concentrations for the sphere-within-a-sphere geometry.

ULTIMATE FINENESS O F A DISPERSION

a(o) = a,, is much closer to the binodal than is the concentration at the outside boundary, a(R) =a,. Figure 2 plots 1 -a l versus a, for the case of y = 0. In the rectangular geometry studied by Nauman and Balsara (1988), this plot would be a 45" line with a, = 1 - a". In the spherical geometry, concentrations are highly skewed. For example, when a, approximates the upper binodal to 7 significant figures (a, = 0.9292798), then a, = 0.2440 compared to a lower binodal con- centration of 0.0707. Spherical systems must be quite large, R > 100fi, in order to have a, reasonably approximate the lower binodal concentration. In a real system, the concentration within a dispersed phase will be relatively closer to one of the binodals than will be the concentration of the continuous phase to the other binodal. This concentration skewing is potentially measurable since fi will be on the order of a few angstroms for small molecules and will be a few hundred angstroms for polymer molecules.

For a fixed a,, there is a range of y for which solutions exist. Values of y near the extremes correspond to physically large systems (large W, small KIR'). The interface is relatively sharp, and both a, and y must be specified to high precision. For intermediate values of y, calculations are numerically simple; but it soon emerges that there is a minimum W (maximum KIR') below which solutions do not exist. This minimum W corresponds to the minimum bifurcation size.

1

0.9

< 0.8 7

i 0 0.7 - I- a

0.6 I-

5 0 0.5 z 0 0 0.4 W n

0.3 I- 3 0

0.2

0.1

0

-

-

-

-

-

- -

-

-

I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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66 DR E.B. NAUMAN et al.

MINIMUM BIFURCATION SIZE

Inherent in the Landau-Ginzburg functional is a compromise between the macroscopic free energy, g , and the gradient energy, ( ~ / 2 ) ( V a ) ~ . A sharp phase boundary at the binodal concentrations minimizes g but maximizes ( ~ / 2 ) ( V a ) ~ . Without phase separation, g is maximized but ( ~ / 2 ) ( V a ) ~ is minimized. For a one dimensional system in rectangular coordinates, Nauman and Balsara (1988) showed that phase separation will only occur if

This result, applicable to systems with d inside the spinodal where gV(ii) < 0, is identical to the minimum size for growth predicted by spinodal decomposition theory (Cahn and Hilliard, 1958; Nauman and Balsara, 1988).

Turn now to the spherical drop goemetry and consider the case of a smooth transition from two phases to one phase as R decreases. For systems slightly above the minimum size, a ( z ) will be nearly flat at d , with a, just above ii and a, below d. Since a ( r ) =d , we approximate &(a) by a first order Taylor expansion

Equation (6) becomes

where K is a constant and B2 = -gU(6). Note that -gV(d) > 0 when ii is inside the spinodal. A solution to Eq. (12) is

K C , sin Bw C, cos pw

a=,+ +

W W (13)

The boundary condition that daldw = 0 at w = 0 gives Cz = 0. The secondary boundary condition is that daldw = 0 at w = W. Imposing this gives

The case of C , = 0 gives

This is the case of no phase separation and a(r ) = constant = d. Equation (14) then forces y = g'(d). For C , # 0 it must be that

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ULTIMATE FINENESS O F A DISPERSION 67

where z = /3W = /3R/fi . The smallest z > O for which Eq. (17) is satisfied is z = 4.4934. Thus, for the sphere-within-a-sphere geometry, bifurcation requires

Comparing Eqs. (10) and (18) gives R, = 1.4303 LC. The inflection point occurs at d2a/dr2 = 0. Equation (13) predicts

(2 - zf) tan zi = 22, (19)

The solution is zi = 2.0816 or R,/R, = 0.4633. We regard the inflection point as the effective boundary between the two phases:

These results for R, and Ri can be confirmed by numerical solution of Eqs. (8) with a, =a. See Case 111 in Table I for a regular solution with x = 3, g"(0.5) = 2 and ~ f i = 3.177.

Equations (18) and (20) rigorously apply only to phase equilibrium within spherical cavities. However, a metastable, multiparticle system could be created by suspending equilibrated drops in a pool of fluid having uniform composition a,. Equation (20) gives the smallest size of a suspended phase that can exist in this form of metastable equilibrium. Smaller drops could presumably be made, say by mechanical action, but would quickly dissolve in the continuous phase. We speculate that this smallest drop size is also the smallest size that will grow by spinodal decomposition. The one dimensional analysis by Nauman and Balsara (1988) gave a minimum bifurcation size which is exactly equal to the minimum size for growth predicted by Cahn and Hilliard (1958). Appendix 2 gives a simplified three dimensional analysis which shows that the minimum bifurcation volume is similar to the minimum size for growth by spinodal decomposition.

Table I shows calculated values of Ri/Rc for various values of the system average composition, d. Note that the analysis applies only for d within the spinodal region where gU(d) < 0. The inflection radius is infinite at the spinodal boundary, but is a weak function of x and B except very close to the boundary. A

TABLE I

Minimum dispersion sizes, RJR,, for binary polymer mixtures

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68 DR E.B. NAUMAN et al.

typical pair of imisable polymers will have x -50 so that Ri ;.: R, is a reasonable approximation for most blend ratios. A typical, high molecular weight polymer has a radius of gyration of about 0.02 microns so the minimum particle diameter is around 0.04 microns.

SURFACE TENSION

The Landau-Ginzburg functional can be used to predict the surface tension between two liquids at equilibrium. Cahn and Hilliard (1958) calculated o for the flat interface in a one dimensional, rectangular system as

where g, = g(a,) and g, = g(a,) are the free energies per unit volume at the upper and lower binodal concentrations and where V, and V, are the volumes the upper and lower phases would have if the interfacial boundary were sharp.

The concept of surface tension is equally applicable to systems of finite extent. For a one dimensional, flat interface,

where L,, and L, are the lengths that the upper and lower phases would occupy given a sharp interface and where L = L, + L, is the total system length. Now, with L finite, a, and a, no longer correspond to the binodal concentrations but are interior to them. If L < LC, then a, = a , = d and the surface tension is zero.

The concentration profile in Eq. (23) is calculated using the rectangular equivalent of Eq. (6 ) ;

Using this rectangular coordinate version of the Euler-Lagrange equation, it can be shown that

where

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ULTIMATE FINENESS O F A DISPERSION

It is also true that

Equations (25)-(27) were derived by Cahn and Hilliard (1958) and by Khacha- turyan and Suris (1968) for the case of L = m. They remain true for finite L, but a, and a, must now be interpreted as extreme concentrations which are necessarily interior to the binodal concentrations. Means for calculating a, and a, are given in Nauman and Balsara (1988). Figure 3 gives results for a regular solution with x = 3. Note that values for a/f i must be multiplied by fi (units of length) and then by pR,T to obtain a in normal units of force/length.

We turn now to the case of a curved interface. Equation (21) still holds, but Eq. (22) must be modified to reflect the geometry. For the case of a sphere-within-a-sphere,

GRADIENT ENERGY PARAMETER, K/R' or K/L' FIGURE 3 Surface tension of Landau-Ginzburg fluids in finite geometries: d = 0.5. x = 3.

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70 DR E.B. NAUMAN er al.

The Euler-Lagrange equation has the form of Eq. (6) , and concentration profiles can be calculated using Eqs. (8). Figure 3 shows results for u / f i as a function of K / R * . There is no simple analog of Eqs. (25) and (27) for the spherical coordinate system.

EQUILIBRIUM PRESSURE PROFILES

The pressure inside a spherical drop with a sharp interface is

The pressure profile is assumed to be flat with P = Po, r < Ri and P = 0, r > Ri. Such a sudden change in pressure is unrealistic for small drops or from any drop when the scale of scrutiny is commensurate with the interfacial thickness. Applying the gradient expression for surface tension to a different form of the Young-Laplace equation gives

z 0 0.5 - r l l i 7 =7.26 -. - I- I

I

I

I LL I

0.3 - I I I I

I omposition, a(r) : 0.1 - 1

I

Pressure, P/P,., 0 - I

I I

-1 I I I I I I I I 0 1 2 3 4 5 6 7

DIMENSIONLESS RADIUS, r l f i

FIGURE 4 Composition and pressure distributions within a spherical drop of a Landau-Ginzburg liquid: ci = 0.015, rc/R2 = 0.019, x = 3.

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ULTIMATE FINENESS O F A DISPERSION

TABLE 11

Characteristics of drops at equilibrium

Case I Case I1 Case 111 Small Large Small Rich Rich Dilute Drop Drop Drop

System Average Concentration, ri 0.150 0.654 0.500 Maximum Concentration, a, 0.929 0.929 0.550 Minimum Concentration, a, 0.119 0.244 0.485 Concentration at Inflection Point, a, 0.550 0.555 0.519 System Radius, W 7.26 8.55 3.18 Fractional Radius at a =a;, W(ai)lW 0.278 0.850 0.465 Fractional Radius at a =0.5 , W(0.5)lW 0.296 0.867 0.657 Fractional Radius at a = ri, W(cS)IW 0.515 0.821 0.687 Surface Tension, a/& 0.363 0.203 0.0022 Maximum Pressure, Po 0.284 0.0561 0.0027 Pressure Calculated as 2a lV . 0.360 0.0557 0.0030 Pressure Calculated as ZolW(O.5) 0.195 0.0547 0.0021 Pressure Calculated as ZolW(ri) 0.338 0.0577 0.0020 K/R' 0.0190 0.0137 0.0989 X 3 3 3

where R , and R, are the principal radii of curvature of the surface of constant concentration and where z is normal to the concentration surface. For the sphere-in-sphere geometry this becomes

The boundary condition associated with Eq. (31) is P = 0 at r = R. Figure 4 shows the calculated pressure profile for a spherical drop at

equilibrium. The concentration profile was obtained by integrating Eqs. (8) for the case of a regular solution with x = 3, d = 0.15 and KIR' = 0.019. Table I1 gives additional characteristics for this and other equilibrium drops. The pressure in this table should be multiplied by pR,T to obtain pressures in conventional units.

CONCLUSIONS

The monotonic composition distributions, calculated here for a spherical coordin- ate system, are qualitatively similar to those determined by Nauman and Balsara (1988). An important difference is that the minimum bifurcation size now corresponds to a finite volume:

This minimum volume represents the smallest drop that can exist in a two phase system at equilibrium. The smallest possible size can represent an important

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72 DR E.B. NAUMAN et al.

physical limit in polymer systems where K is large or in confined structures, such as molecular sieves, where the physical dimensions are small.

Another difference between the current results and those of Nauman and Balsara (1988) is the inherent asymmetry of the sphere-within-a-sphere geometry. Concentrations at the center are closer to one binodal than concentrations at the outer wall are to the other binodal. It is speculated that the resultant skewing in concentrations may be measurable. Numerical integration of Eq. (8) is difficult for small values of KIR', and an improved algorithm for estimating a( r ) in large systems is desirable.

The surface tension has been calculated for finite systems with both flat and curved interfaces. It vanishes for systems below the critical bifurcation size. In larger systems with curved interfaces, it gives rise to a pressure gradient according to Eq. (30). The internal pressure agrees with the classical result

for large drops. The classical theory overpredicts the internal pressure for small drops, but this overprediction is partially due to the inherent ambiguity in defining the radius of the drop. We prefer the inflection radius, Ri, where d2a/dr2 = 0, for most purposes. However, use of the radii where a = r f or a = 0.5 can be argued and sometimes give better approximations for Po.

REFERENCES

Ariyapadi, M.V., and Nauman, E .B . , "Gradient Energy Parameters for Polymer-Polymer-Solvent Systems and Their Application to Spinodal Decomposition in True Ternary Systems", to appear I . Poly. Sci., Part 8, Poly. Phys., 1990.

Cahn, J.W., and Hilliard, J.E. , "Free Energy in a Nonuniform System", I . Chem. Phys., 28, 258 (1958).

Khachaturyan, A.G. , and Suris, R.A. , "Theory of Periodic Distribution of Concentrations in a Supersaturated Solid Solution", Sovier Physics-Crysfallography, l3, 63 (1968).

Nauman. E.B., and Balsara, N.P. , "Phase Equilibria and the Landau-Ginzburg Functional", Fluid Phase Equilibria, 45,229 (1988).

APPENDIX 1

THE GENERALIZED CHEMICAL POTENTIAL

Consider a classical binary mixture with K = 0. The Gibbs free energy, g, and the chemical potentials, pA and p,, satisfy the following relationships:

(i) g is a function of state for systems at equilibrium (ii) The chemical potentials can be defined as

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ULTIMATE FINENESS O F A DISPERSION 73

(iii) The following equations are satisfied

(iv) pA and pB are constant at equilibrium. Turn now to the case of K > 0 and define a new thermodynamic function,

where a, represents a concentration at some arbitrary point within the system. We first show that T is a function of state at equilibrium. The Euler-Lagrange equation can be integrated with respect to a to give

The left side of Eq. (A6) is a function of state at equilibrium. Then so must be the right side and it follows that T, like g , satisfies condition (i).

We now apply condition (ii) to the new function T to define generalized chemical potentials for the case of K > 0:

Condition (iii) is satisfied automatically (with T replacing g ) by this definition. To show that Condition (iv) is satisfied, rewrite Eq. (A6) as

where C = g(a,) - a , y is a constant. The Euler-Lagrange equation is

Substituting Eqs. (A9) and (A10) into Eqs. (A7) and (A8) gives

so that the generalized chemical potentials are constant at equilibrium. Note that the specific values for p; and p; depend on the choice for a,. Whatever the choice, p; and pk are system constants at equilibrium.

The integral in Eq. (A5) is a function of state only at equilibrium and is path dependent in nonequilibrium situations. Thus the generalized chemical potentials are defined only under equilibrium conditions. However, the gradients of the chemical potentials are defined under nonequilibrium conditions and can be used

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74 DR E.B. NAUMAN er al.

as the driving force for diffusion:

J = J A = - D . V p A = - J B = + D ( l - a ) V p B ('413)

where J is the diffusive flux and D is the molecular diffusivity. Equation (A13) can be written as

J = ( l - a ) J + a J = ( l - a ) J A - a J 8

= - ( I - a )Da V p A + a D ( 1 - a ) V p ,

= - D a ( l - a ) V ( p A - p,) (A141

The corresponding continuity equation is

where the p, - p, term was evaluated using Eqs. ( A 7 ) and ( A S ) . The above results are valid in three dimensions for arbitrary coordinate

geometries. They represent a generalization of the results of Nauman and Balsara (1988) who, although they used the V notation, obtained some results which apply only to one dimensional systems. Specifically, their results for pa and pl, used an analytical integration of Eq. ( A 6 ) which is valid only for the one dimensional case.

APPENDIX 2

ANALYSIS O F MINIMUM DROP SIZE

Equation (A15) can be written as

Linearization gives

A three dimensional version of this equation is

where D = i ( 1 - i ) D . Consider a solution of the form

a = i + ~e~ sin ax sin by sin yz

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ULTIMATE FINENESS O F A DISPERSION 75

For Eq. (A19) to be a solution to Eq. (AH), it must be that

For the disturbance to grow, it must be that I > O . Thus the minimum size for growth is given

If we set p2 = y2 = 0, we obtain the result of Cahn and Hilliard (1958).

which is consistent with Eq. (10). Thus the one dimensional, equilibrium theory of Nauman and Balsara (1988) predicts that the minimum bifurcation size is equal to the minimum size for growth predicted by the dynamic theory of Cahn and HIlliard (1958).

For the three dimensional case, a separating phase will have volume

This volume is minimized-subject to the constraint of Eq. (A21)-when cu = /3 = y, a result which also agrees with notions of isotropy. The separating phases form cubes with sides of length

so that L, is larger by a factor of fi than for the one dimensional case. For the volume we take two cubes to form a complete phase structure consisting of both lean and rich portions:

The volume for the sphere-within-sphere geometry is

These volumes are comparable given the approximate nature of the analysis.

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